Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be false

Question

What are some examples of serious mathematical theory-building around hypotheses that are believed or known to be false?

One interesting example, and the impetus for this question, is work in number theory based on the assumption that Siegel zeros exist. If there were such things, then the Generalized Riemann Hypothesis would be false, which it presumably isn't. So it's unlikely that there are Siegel zeros. Still, lots of effort has gone into exploring the consequences of their existence, which have turned out to be numerous, interesting, surprising and so far self-consistent. The phenomena generated by the Siegel zero hypothesis are sometimes referred to as an "illusory world" or "parallel universe" sitting alongside that of ordinary number theory. (There's some further MO discussion e.g. here and here.)

I'd like to hear about other examples like this. I'd be particularly grateful for references, especially those that discuss the motivations behind and benefits of undertaking such studies. I should clarify that I'm mainly interested in "illusory worlds" built on hypotheses that were believed to be false all along, rather than those which were originally believed true or plausible and only came to be disbelieved after the theory-building was done.

Further context: I'm a philosopher interested in counterfactual reasoning in mathematics. I'd like to better understand how, when and why mathematicians engage with counterfactual scenarios, especially those that are taken seriously for research purposes and whose study is viewed as useful and interesting. But I'd like to think this question might be stimulating for the MO broader community.

Answer 1

Girolamo Saccheri in his Euclides Vindicatus (1733) essentially discovered Hyperbolic Geometry, by building around the hypothesis that the angles of a triangle add up less than 180°. This was widely believed to be always impossible, since people at that time were convinced of the absolute nature of Eucliden Geometry.

Answer 2

Computational complexity theory involves investigating illusory worlds, since so many of the results depend on unanswered questions. A vivid example is given by Russell Impagliazzo's paper "A Personal View of Average-Case Complexity" where he discusses different hypotheses related to P vs. NP. He describes it in terms of 5 possible worlds, which he gives colorful names: Algorithmica, Heuristica, Pessiland, Minicrypt, Cryptomania. By necessity, at most one of these worlds can be true, so the other 4 are illusory. He discusses the implications for computer algorithms in each possible world. Algorithmica is the world in which P = NP, but the other four worlds consider different ways in which P != NP, which have different consequences for applications such as cryptography.

Answer 3

The first mathematical objects studied that are believed not to exist seems to be odd perfect numbers

In 1496, Jacques Lefèvre stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists.

Euler began the study of the properties of odd perfect numbers, showing any such number must be of the form $q^\alpha N^2$ with $q\equiv 1\equiv \alpha \bmod 4$. Many later results are on the linked Wikipedia page. They are all properties of the numbers themselves, however, and the existence of an odd perfect number does not seem to have consequences elsewhere, unlike the Siegel zero.

Answer 4

I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he instead built the beautiful and elaborate theory, now much studied and extended.

Many set theorists today do not view this as an instance of counterfactual reasoning, since they think measurable cardinals are consistent with ZFC, but from Silver's point of view, he was developing the elaborate theory in attempt to refute the measurability assumption.

Of course, in light of the incompleteness theorem, we know that Silver's view is at least as consistent as his opposition, since if ZFC is consistent, then it is consistent with ZFC to suppose that measurable cardinals are not consistent. So one cannot really criticise Silver's view as incoherent.

Answer 5

I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately as true.

Examples would include:

• The axiom of constructibility $V=L$. This is the hypothesis introduced by Kurt Gödel in order to prove the relative consistency with ZF of the axiom of choice and the continuum hypothesis. There are hundreds if not thousands of published papers developing the nature of set theory under this hypothesis, but at the same time, it is a standard view in set theory, especially large cardinal set theory, that this axiom is not ultimately "true" of the intended platonic set-theoretic realm. Maddy has written on this, explaining how it violates her maximization maxim. Meanwhile, I have argued that a multiverse perspective allows for a more forgiving perspective on this axiom (See Hamkins, Joel David, A multiverse perspective on the axiom of constructibility, Infinity and truth. LNS, NUS 25, 25-45 (2014). ZBL1321.03061.)

• The inner model theories of large cardinals. This is an extremely active subject in set theory, developing the analogue of the constructible universe for the large cardinal context, with again hundreds of researchers and many papers, developing the intricate theories of these models. And yet, the most common standard view amongst large cardinal set theorists, even those undertaking the inner model theory, is that the actual (platonic) set-theoretic universe is not actually one of these canonical inner models. We study them, to learn about what things would be like in these inner models, since this allows us to prove relative consistency results, and gives us a deeper understanding of the large cardinal hypotheses in question.

• In some parts of the inner-model analysis, the inner model theories are developed under an anti-large cardinal assumption, that there is no inner model with a certain kind of large cardinal. This assumption is not held ultimately as true, but it can be true in inner models, and can be viewed as a kind of inductive analysis.

• The axiom $V=L(\mathbb{R})$, intensely studied in the context of determinacy. Set theorists study this theory especially under the assumption of AD+DC. Again, it is not part of the set-theoretic conception that this axiom is true, but rather only that it is true in that inner model, where the determinacy issues have certain very useful consequences.

In all these cases, set theorists have introduced and developed an ongoing elaborate foundational theory, which for philosophical reasons is not viewed ultimately as true.

Of course, a fundamental philosophical issue here is the difficulty of saying what it means for a foundational set theory to be "true."

Answer 6

This is more of a theoretical/mathematical physics example, but it can happen in mathematical physics that a lot of stuff is built around hypotheses which are essentially known to be false from the beginning or objects which are known not to exist. Some examples:

1.) I know that you asked for hypotheses which were always known to be false, but Tait originally started to develop knot theory because Kelvin had hypothesised that atoms can be obtained as knots in the ether. The concept of the ether was invalidated by experiments of Michelson and Morley which obviously also invalidated the hypothesis of Kelvin regarding the physical basis of knot theory. The mathematics of knots has survived and flourished up to the present day although this application did not work.

2.) A huge amount of theoretical and mathematical papers have been published on magnetic monopoles (including Dirac and 't Hooft-Polyakov monopoles) although the consensus looks to be that magnetic monopoles likely do not exist in this Universe which we live in. I am not sure if Dirac regarded the hypothesis of existence of monopoles to be false ''all along'', but this is certainly possible, as magnetic monopoles are forbidden by the mathematical equations of classical electromagnetism and Dirac was considering what happened theoretically if you increased the amount of symmetry which the equations have.